Compact actions whose orbit equivalence relations are not profinite

Abstract

Let Γ↷(X,μ) be a measure preserving action of a countable group Γ on a standard probability space (X,μ). We prove that if the action Γ↷X is not profinite and satisfies a certain spectral gap condition, then there does not exist a countable-to-one Borel homomorphism from its orbit equivalence relation to the orbit equivalence relation of any modular action (i.e., an inverse limit of actions on countable sets). As a consequence, we show that if Γ is a countable dense subgroup of a compact non-profinite group G such that the left translation action Γ↷G has spectral gap, then Γ↷G is antimodular and not orbit equivalent to any, not necessarily free, profinite action. This provides the first such examples of compact actions, partially answering a question of Kechris and answering a question of Tsankov.